Comparing models of microbial-substrate interactions and their response to warming

7 Recent developments in modelling soil organic carbon decomposition include the explicit 8 incorporation of enzyme and microbial dynamics. A characteristic of these models is a 9 positive feedback between substrate and consumers, which is absent in traditional first order 10 decay models. Under sufficiently large substrate, this feedback allows an unconstrained 11 growth of microbial biomass. We explore mechanisms that curb unrestricted microbial growth 12 by including finite potential sites where enzymes can bind and by allowing microbial 13 scavenging for enzymes. We further developed a model where enzyme synthesis is not scaled 14 to microbial biomass, but associated with a respiratory cost and microbial population adjusts 15 enzyme production in order to optimise their growth. We then tested short and long-term 16 responses of these models to a step increase in temperature, and find that these models differ 17 in the long-term, when short-term responses are harmonized. Oscillations that arise from a 18 positive feedback between microbial biomass and depolymerisation are eliminated if 19 limitations other than through enzyme-substrate interactions are considered. The model, 20 where enzyme production is optimised to yield maximum microbial growth shows the 21 strongest reduction of soil organic carbon in response to warming, and the trajectory of soil 22 carbon largely follows that of a first order decomposition model. Modifications to separate 23 2 growth and maintenance respiration generally yield short-term differences, but results 1 converge over time, because microbial biomass approaches a quasi-equilibrium with the new 2 conditions of carbon supply and temperature. 3 4


Introduction
Traditional soil organic matter decomposition models are based on first order kinetics, where 10 decomposition scales to the pool size. The scaling factor represents recalcitrance of a specific 11 pool and is modified by soil temperature, moisture, and other soil properties (e.g., van Veen et  14 dissolved organic carbon (DOC) by extracellular enzymes. This depolymerisation step is thought 15 to be a rate-limiting step in organic matter decomposition processes (Schimel and Weintraub,16 2003; Fontaine and Barot, 2005). 17 In traditional models, microbes are only considered as a simple donor-controlled pool (i.e., 18 microbial biomass has no impact on decomposition), or in an implicit manner (Gerber et al., 19 2010). In contrast, in microbial models, decomposition rates become a function of enzyme 20 activity that is linked to microbial biomass (Allison et al., 2010;German et al., 2012). This leads 21 to more complex dynamics because decomposers feed back into soil organic matter degradation 22 via microbial enzyme production affecting depolymerisation. This positive feedback between 2 Materials and methods 8 9 We first introduce three model families that differ in the way depolymerisation is handled. 10 In all models, the setup consists of a single soil organic matter pool and a single microbial pool 11 ( Fig. 1). All models also implicitly take into account interaction between enzymes and substrate 12 that results into depolymerisation of substrate into a DOC pool on which microbes can feed. 13 Enzyme-substrate reactions are based on Michaelis-Menten kinetics (see Appendix A, 14 Michaelis-Menten kinetics with enzyme denaturation). We do not consider a specific enzyme 15 pool, nor a specific DOC pool, but assume that the enzyme and DOC pools are in a quasi-steady 16 state (see Appendix A, DOC and enzyme dynamics). Thus, the amount of enzyme produced 17 equals the amount of enzyme decay at every time step. Similarly, the amount of DOC produced 18 is the same as the amount of DOC consumed by microbes. In contrast to Allison et al. (2010), 19 but congruent with German et al. (2012), there is no "free" DOC, both fresh litter and microbial 20 necromass need to be depolymerised before they can be ingested by microbes. In all models 21 depolymerisation and microbial respiration are temperature dependent, causing increased 1 depolymerisation and reduced microbial CUE with warming. (2) 7 where S and M are the soil organic matter and the microbial pool, respectively, I is the input of 8 fresh litter, λ d is the death rate of microbes, D is the rate of depolymerisation, and ε is the 9 microbial CUE. 11 In the forward model (FWD), depolymerisation is represented as a Michaelis-Menten process 12 and stems from the simple microbial-enzyme decomposition model as proposed by Allison et al. 13 (2010) and modified by German et al. (2012) (Fig 1a). (3) 15 Where D is the rate of depolymerisation, V max,FWD is the maximum depolymerisation rate and K m 16 the half saturation constant of enzymes. Appendix A shows the derivation of this function based 17 on enzyme-substrate dynamics. (4) 6 Where V max,REV is the maximum depolymerisation rate for this model, K e is a half saturation 7 constant that determines the diminishing return function. In the cases developed in the Appendix, 8 K e incorporates factors indicating the finite sites for enzyme substrate interactions (Appendix B, 9 model with limited available substrate), or the efficiency with which microbes scavenge for free 10 extracellular enzymes (Appendix B, microbial consumption of enzymes). A version of the 11 reverse Michaelis-Menten model also has been derived for the case where an enzyme can adsorb 12 to only a fraction of soil organic matter due to inaccessible binding sites from surface limitation 13 or phyiscal protection (Wang and Post, 2013). A major difference from the FWD model is the 14 inclusion of the amount of microbial biomass in the denominator in lieu of soil organic matter. 15 Therefore, the depolymerisation per unit biomass decreases as biomass increases, plateauing at 16 V max,REV *S (diminishing return). 18 In our OPT model, we relax the condition that microbial enzyme production scales to microbial 19 biomass, an assumption that is present in many microbial models and which is also assumed in 20 the FWD and the REV model above. Instead, we probe a model where microbial enzyme 21 production is optimised for growth. We motivate this by microbial competition (Allison, 2005), 22 which allows microbes to succeed if microbial enzyme production allows the highest possible 1 return. Optimisation only has meaningful results for the case of limited substrate availability (i.e. 2 a diminishing return, possibly through constraints in potential sites for enzyme-substrate 3 reaction) and if there is a cost associated with microbial enzyme production.

4
Depolymerisation as a function of enzyme production can be represented by V max,OPT is the maximum rate of depolymerisation, P is the enzyme production rate, and K p 7 carries information on the affinity of the enzyme for the substrate and longevity of the enzyme 8 (see Appendix C, for full derivation of depolymerisation in the OPT model). 9 Microbial growth (G) is as in previous models but accounts for carbon expenditure of enzyme 10 production: Where c is the respiratory cost per unit enzyme produced (Schimel and Weintraub, 2003). 13 Optimising growth by setting dG dP = 0 yields: 15 And the cost per unit carbon depolymerised is then:  18 While the previous models are fairly simple, we further reduce the complexity by removing 19 microbial biomass as a state variable but instead consider M at a quasi-steady state (QSS). In the QSS microbe models, the microbial uptake at each time step is thus equal to the microbial carbon 1 loss via death or respiration (Fig 1b). This is identical to our treatment of DOC and enzymes, 2 where production and removal of these substances are always balanced. This simplification is 3 motivated by the fact that microbial biomass turns over much faster than soil organic matter, and 4 therefore microbial biomass adjusts much faster to changes in environmental conditions than soil 5 organic matter itself. The fast turnover of M compared to S allows microbial biomass to (quasi)-

Quasi-steady state (QSS) microbe models
In turn, depolymerisation is immediately partitioned into respiration and a returning carbon flux, 20 which mimics microbial death.  22 While the dynamics of the soil organic matter pool remains the same as in the base model setup, 1 we alter all models (FWD, REV, OPT) to treat growth and maintenance respiration as separate 2 processes (Fig 1c). Partitioning of microbial respiration into growth and maintenance respiration 3 characterise the microbial pool as follows:

Partitioning between maintenance and growth respiration
Where g is the growth respiration fraction and λ r the maintenance respiration rate. The separation 6 of microbial respiration in growth and maintenance terms is motivated by a similar formulation   14 The last model represents the structure of traditional decomposition models such as CENTURY 15 (Parton et al., 1987) or Roth-C (Coleman et al., 1996) and their derivatives, where decomposition 16 is considered as a first-order reaction: 18 where k is the first order decomposition constant. The two major differences between our first- 19 order decomposition (FOD) model and traditional models are that we consider only a single 20 carbon pool whereas traditional models consider multiple pools with different turnover times that 21 feed into each other. We also consider a temperature-dependent CUE on top of a temperature-dependent processing rate (k, see parameterisation and implementation section). This increases 1 the fraction of carbon processed with warming to become CO 2 . Respiration (R) is then

First-Order Decomposition (FOD) Model
3 2.2 Temperature response 4 We implement the response of decomposition to warming by modifying the depolymerisation 5 and the microbial respiration. 6 In the FWD, REV and OPT model, V max is modified as Where V max,i and V max,i (ΔT) are the reference and temperature-dependent maximum 9 depolymerisation rate of the model i = (FWD, REV, OPT, see Table 3). Similarly, the 10 decomposition rate k is modified by the Q 10 function in the FOD model.

11
Further, we also parameterise CUE as a linear function of the temperature change, following

18
Where λ r,0 and λ r (ΔT) are maintenance respiration rate at reference and elevated temperature. 19 Growth respiration is typically much less sensitive to warming than maintenance respiration  organic carbon and the same initial microbial biomass. Both CUE (ε), and its temperature 10 dependence (ε slope ) are the same across models. Further, the temperature sensitivities of V max 11 are identical across models so that we obtain the same increase of depolymerisation in the first 12 time step after the temperature perturbation. We motivate this kind of parameterisation by 13 acknowledging that many of these parameters are largely unknown, but it will provide us with 14 the possibility of comparing the functional response to long-term warming across these models. 15 We use parameters as reported in German et al. (2012), with a few modifications. Here, we 16 report V max,FWD and K m by considering 15°C as our reference temperature and by incorporating 17 German et al. (2012) tuning coefficients (a K , a V ) directly into these two parameters ( determination of V max,REV , which is tuned here to such that the REV model yields equivalent 10 equilibrium values of S at the reference temperature as the FWD model.

11
In the OPT model, we adjust V max,OPT (in the same manner as in the REV model) such that the 12 system again yields equilibrium values for S at the reference temperature (15°C) and the same 13 initial response to warming as in the other models. In the OPT model, we have to work with two 14 additional parameters, namely the cost of enzyme production (c), and the term that contains the 15 affinity of enzymes for the substrate (K p ). We chose to have the OPT models comparable to 16 others if the cost (c) is zero. Higher costs (c>0) therefore, will yield different equilibrium result 17 of S and a different response to warming, depending on the cost of enzyme production. 18 Both, the half saturation constant (affinity parameter, K p ) and the cost per enzyme produced are 19 parameters that are hard to come by. Instead, the relationship between enzyme production cost 20 and overall depolymerisation allows us to quantify the product of K p and c. (see Eq. 8 in the 21 main text). We define a fractional expense μ that quantifies the enzyme expenditures relative to We chose μ to be 0, 10, and 50 percent of the depolymerisation rate at the reference temperature 1 and at steady state. Based on the relationship given in Eq. 8 we then obtain an expression for the 2 combined cost (c) and the half saturation constant (K p ) without having to specify the value of the 3 individual parameters (see also the variable Y in Table 2): 4 K p * c = μ 2 * D Eq.,ΔT=0 5 Where D Eq.,ΔT=0 is the rate of depolymerisation at zero enzyme cost and reference temperature. 6 When separating growth and maintenance respiration, we sought to equalise steady state CUE, 7 M, and S by tuning g and λ r . We first parameterised maintenance respiration, where, the 8 coefficient for maintenance respiration is scaled to microbial turnover (Van Bodegom, 2007). 9 We motivate the partitioning between growth and maintenance respiration based on vegetation The second parameter, g is adjusted, such that the CUE at the steady state and reference 16 temperature remains the same. This constrains g to 18 To obtain the same equilibrium values of CUE at 20°C as in the base models, we adjust Q 10, r

19
such that models with maintenance respiration have the same CUE as the base models. 20 Finally, in the FOD model, the traditional decomposition model, we adjust the parameters k and 1 ε 0 to obtain the same S, and CUE as in all other models at 15°C and employ a Q 10,k value 2 identical to the Q 10 values of V max in the other models. We keep the decreasing CUEa feature 3 not typically set up in traditional models.

4
All parameter values are given in Table 3. parameter adjustments. Also, by identical Q 10 of V max and CUE's, the initial response to a 12 warming is equal across the models. 13 In all models, warming leads to a decline of soil organic matter and microbial biomass (Fig. 2).
14 In this initial comparison, we assume that there is no cost associated with microbial enzyme 15 production. Across all the models, microbial biomass first increases because of higher 16 depolymerisation. Increased depolymerisation causes soil organic matter to decrease. In the 17 longer term, M decreases as rates of depolymerisation decline due to a reduction in S, and due to 18 lower CUE. We note that M becomes identical across all models in the long term when soil 19 organic carbon has equilibrated with microbial processing at higher temperature (see also Table   20 2). warming triggers an increase in depolymerisation, which in turn feeds microbial biomass, 2 causing an even higher rate of depolymerisation. This positive feedback experiences a break only 3 when the substrate (S) is sufficiently depleted, such that microbial biomass begins to decline.

4
Thereafter, the positive feedback takes over again, the decreasing microbial biomass spirals 5 down along with depolymerisation until microbial biomass is low enough for soil organic matter 6 to recover. The amplitude of the oscillations dampens over time (Fig. 2). Rates of respiration 7 oscillate along with microbial biomass, before settling at the initial rate in the long term (after ca. 8 200 years). 9 The transient dynamics in the REV model with a diminishing return as enzyme (or microbial) 10 concentration increases are smoother compared to FWD model (Fig. 2). The mechanism of 11 allowing a finite site for enzyme-substrate reaction or microbial scavenging for enzymes curbs 12 the growth of microbial biomass. Warming still leads to an initial increase of microbial biomass, 13 owing to the fact that the gains of depolymerisation outweigh losses from increased respiration 14 (i.e. decreased CUE). As soil organic matter depletes, microbial biomass is reduced, ultimately 15 below the initial levels. 16 The OPT model considers the metabolic cost of enzyme production and allows optimisation of  The analysis of equilibria helps to understand the model behaviour. We first address the "long 2 time scale" in Table 2 where we solve for the steady state of the entire system (i.e.  model. In the OPT model the resulting equilibria of S and M end up being complex expressions, 10 and we did not calculate the long-term equilibria of M but expressed them simply as a function 11 of soil organic matter. Further, the steady states of S are the same in the traditional first order 12 model (FOD) and the OPT model with zero cost. As expected, the effect of enzyme production 13 cost has a negative impact on microbial biomass. 14 The analysis of the short-term quasi-steady state of the microbial biomass ( dM dt = 0) is useful to 15 understand the trajectory of the coupled S-M system. Typically, microbial turnover is much 16 faster than the turnover of bulk soil organic matter (Stark and Hart, 1997; Schmidt et al., 2007). 17 Thus, we would expect that microbial biomass is approaching a quasi-steady state given any 18 level of S. 19 In the FWD model, we find that the quasi-steady state for M requires a perfect balance of 20 parameters that govern growth-and death rates ( higher enzyme production cost act to reduce M ̅ in these models. Given the quasi-equilibrium biomass, and the resulting decomposition at quasi-steady state, we 10 set up a second line of modelling experiments, where depolymerisation rates, as well as 11 microbial respiration and death, are calculated based on microbial biomass at quasi-steady state 12 (QSS microbe, Table 2, second and third columns, see also method section 2.1.2). Compared to 13 the base models, the QSS-microbe models yield very similar results for S and respiration, but values of the base models in REV and the OPT model, and therefore, the quasi-steady state 18 appears to be an acceptable assumption over medium to long time scales. Our results further 19 show that the depolymerisation in the OPT model at quasi-equilibrium and at marginal enzyme 20 production cost (μ 0) yields a depolymerisation formulation that is functionally the same as a 21 first order decomposition model. Depolymerisation in the OPT model becomes V max *S in 22 absence of enzyme production cost (see Table 2), and therefore, the entire dynamics has the 1 familiar first order characteristics (compare Eqs. 9 and 11). In the third modification of our base models, we partition respiration in our models into a 4 temperature independent growth respiration and a temperature (and biomass) dependent diminishing (REV and OPT). Similar to microbial biomass, differences disappear within <1 year 18 after the step warming. We note that in our modelling setup, we adjusted the temperature 19 sensitivity of the maintenance respiration such that CUE is the same at the reference (15°C) and 20 the elevated (20°C) temperature. Finally, we analyse in the OPT model how levels of costs associated with enzyme production 1 affects soil carbon storage and response to temperature (Fig. 4). Because of largely unknown 2 parameters we express enzyme expenditures as the fraction of respiratory carbon for enzyme 3 production per unit carbon depolymerised at the reference state (see Eq. 8 and Eq. 16). We tested 4 3 levels of enzyme production cost: 0%, 10%, and 50% of equilibrium depolymerisation at our 5 reference condition (i.e. 15°C). As expected, increasing enzyme production cost reduced the rate 6 of depolymerisation, and S is therefore maintained at a higher level. The increasing costs also 7 resulted in a smaller relative decline of S in response to warming, whereas the absolute loss is

17
Many microbial decomposition models work under the assumption that enzyme production is 18 proportional to microbial biomass, however it is also conceivable that microbes are adjusting The response to temperature in our OPT model closely resembles the traditional first order decay 8 model (FOD). In the limit of enzyme production cost approaching zero, depolymerisation occurs 9 at the maximum rate (V max *S), essentially turning the OPT model into a first order model ( Fig.   10 2). In the OPT model, reductions in depolymerisation via K p are alleviated when enzyme 11 synthesis is inexpensive, where the reduction of the maximum depolymerisation rate becomes a 12 function of the product of K p *c (Eq. 7 and Table 2). The results of the OPT model also show the 13 effects on assumptions on microbial enzyme production rates. In many microbial models, 14 enzyme production is scaled to microbial biomass. Lifting the tight coupling between microbial 15 biomass and enzyme prodcition leads to a more dynamic enzyme concentrations and ultimately 16 affects the temperature sensitivity of decomposition. Thus, the cost and trade-offs associated 17 with microbial enzyme production are potential important areas to better quantify the long-term 18 response of soil carbon storage to climate change. 19 The response of decomposition to warming can be viewed as a response occurring on multiple 20 timescales. For example, while enzyme activity likely produces an immediate response, 21 microbial respiration responses may also be triggered quickly, although longer term acclimation 22 may occur (Frey et al., 2013). It may take longer for microbial biomass to respond to temperature 23 changes (weeks to months). Finally, because the rate of decomposition is slow compared to the 1 overall abundance of soil organic matter, discernible changes in this pool occur on timescales of 2 months to years. Based on the distinct rates of adjustments, timescales canin principlebe 3 separated by assuming a quasi-steady state of pools that turn over fast.

4
The assumption that both enzyme concentrations and DOC (i.e. the depolymerisation products) 5 are at quasi-steady state cuts across all models presented here (FWD, REV and OPT, see 6 Appendix A). When we extend our assumption of steady state to the microbial timescale (quasi-7 steady state of microbial biomass), we find that for both the REV and the OPT model, the short-8 term response of microbial biomass and respiration is influenced by the adjustment of microbial 9 dynamics to the warmer temperature (Fig. 3). Because microbial biomass jumps immediately to 10 a higher level after the temperature increase in our QSS assumption, depolymerisation, and thus 11 respiration, are affected. However, the QSS assumption affects the trajectory of the soil carbon 12 pool only minimally. At timescales that allow microbes to turn over a couple of times (several 13 months), the quasi-steady state poses a suitable approximation to represent respiration and 14 microbial biomass, even after a sharp perturbation in the form of a step change. In the QSS 15 assumption, depolymerisation becomes independent of the microbial biomass (but is still 16 dependent on a combination of microbial parameters, see Table 2). 17 The introduction of QSS microbial biomass allows addressing and comparing the long-term 18 responses of the different models to warming. In particular, the comparison of the QSS derived 19 depolymerisation of the FOD with the REV and the OPT directly show the effect of how 20 enzyme-substrate affinity and enzyme production costs dampen the rate of depolymerisation and 21 its response to temperature. In other words, the long-term response of the FOD is equivalent to 22 the long-term response of our OPT or REV model, when 1) K e is low (high enzyme production, 23 high enzyme-substrate affinity, and low enzyme turnover), and/or 2) costs of enzyme 1 productions are low, and 3) and CUE (the fraction of depolymerised not respired but cycled back 2 into soil organic matter pool) is also temperature dependent in the FOD, a feature typically not 3 included in traditional decomposition models.

4
CUE ultimately is the result of different microbial respiration terms. Here, we consider 3 5 processes that may affect microbial respiration under a warming scenario. We first consider a 6 partitioning into growth and maintenance respiration across our 3 models. Growth respiration is 7 simply assumed to be a proportion of carbon allocated to microbial growth. In contrast, 8 maintenance respiration scales to microbial biomass in our models, where the proportionality 9 factor increases with temperature. We motivate the partitioning by formulations of plant 10 respiration in terrestrial biosphere models. We find that this separation affects the short-term  In the OPT model, we introduce an additional respiration term, namely the cost of enzyme 18 production. In this model, we allow microbes to adjust enzyme production in order to optimise 19 growth. It is interesting that increasing costs lead to a smaller immediate response in respiration 20 and more resilient soil organic matter pool in the long term, when subject to warming (Fig. 4). 21 The early respiration response in the OPT model is both a product of higher rates of 22 depolymerisation (increased V max ), but also a higher rate of enzyme production. However, the 23 enhancement relative to the rates at the reference temperature becomes smaller with higher 1 enzyme production cost. In the long term, the decrease in soil organic matter is reduced when 2 enzyme production costs are considered. This reduction is accompanied by a smaller reduction in 3 CUE under higher enzyme production, even though there is a subsequent CUE reduction 4 occurring as S declines. The changing yield tradeoff overall acts to buffer respiration increases 5 that could be expected from physiological responses alone (V max ), although the effects are 6 smaller and may be well within the uncertainty of the temperature response of any parameters 7 considered here. We note that enzyme expenditure relative to depolymerisation is a function of 8 the product of K p and c. 9 We acknowledge that we used a simplified set-up of our model suite. For example, we assumed Since the response to warming is vastly different across our suite of models, 14 our results suggest that the degree of enzyme limitation and the microbial response to enzyme 15 limitation are potential areas that could help constrain the quantification of the long-term 16 response of soil organic matter to warming. (and/or enzyme-substrate affinity is high), and 4) and microbes turn over relatively fast 12 compared to soil organic matter. Our results thus suggest that a better grasp of the limiting steps 13 of decomposition and mechanisms of microbial enzyme production will help to constrain the 14 long-term response to warming. 15 16 Appendix A

17
Michaelis-Menten kinetics with enzyme denaturation 18 The dynamics of the enzyme-substrate complex are  17 We assumed that DOC concentrations are in equilibrium with substrate and microbial uptake. In Because enzyme turn over fast, we can assume a quasi-steady state of the total enzyme pool by 6 setting Eq. A6 to zero. We obtain: And depolymerisation as: [S]+K m (A8) 10 Finally, microbial decomposition models assume that enzyme production is proportional to the With Yet, it is conceivable, that the enzyme-substrate complex, and free enzymes decay at different 15 rates (see also Eqs A1 and A2).  Microbial consumption of enzymes 8 Microbes feeding on free enzymes can be represented as: Where F is microbial enzyme consumption and λ E,M the feeding rate. We can then represent the 11 decay of the free enzymes with 12 [E]* λ E1 = [E]( λ E1,0 + λ E,M *M) (B2) 13 where the total λ E,0 is the spontaneous enzyme decay rate. Model with limited available substrate 5 Access to substrate might be finite, for example, if organic matter is associated with mineral soil 6 or if the rate of depolymerisation is constrained by the surface area. In this case, the relationship 7 between the total available substrate and the free sites can be calculated as Where S f are the available sites for enzyme reaction, θ a scalar relating the total amount of 10 substrate to the total potentially free sites (e.g. a surface to mass conversion), and and with P proportional to microbial biomass (M) Where V max = K cat * b λ E2

10
In this case, depolymerisation and microbial consumption is independent of the substrate but is 11 determined by the relative rate of catalysis and irreversible destruction of the enzyme-substrate 12 complex.
This implies that enzymes mainly decay if they are not associated with the substrate and that 15 there is an appreciable amount of free enzymes. This is realistic under substrate limiting 16 conditions, as there will be a sizeable amount of free enzymes compared to enzyme substrate 17 complexes. 18 We 14 Microbial growth (G) will be 15 G = (1-g) * (D-Pc-λ r *M) (C2) 16 Where g is the growth respiration factor, c the respiratory cost per unit enzyme production, and 17 λ r the maintenance respiration factor. 18 Enzyme production (P) can be optimised by substituting Eq. C1 into Eq. C2 and setting dG dP = 0.
1 This yields: The proportion of carbon expended for enzyme production relative to depolymerisation is Instead of specifying c, we used Eq. C4 to express overall microbial carbon expenditure for 6 enzyme production. After assigning a value to μ, we calculate c based on equilibrium S at 7 reference temperature. 8 In contrast, the microbial scavenging scenario does not provide an optimum enzyme production. 9 In this case, depolymerisation is: (K e +M) * λ E (C5) 11 And thus, dG dP will yield a constant where growth scales with the rate of enzyme production.    1998. 17 Wagai, R., Kishimoto-Mo, A. W., Yonemura, S., Shirato, Y., Hiradate, S., and Yagasaki, Y.: 18 Linking temperature sensitivity of soil organic matter decomposition to its molecular structure, 19 accessibility, and microbial physiology, Glob. Change Biol., 19, 1114-1125,

FWD Model with maintenance respiration
As FWD model but microbial respiration is partitioned into temperature insensitive growth and temperature sensitive maintenance respiration terms.

REV Model
Depolymerisation and uptake relative to microbial biomass decreases with increasing M (diminishing return mechanism).

REV Model with equilibrium microbes
As REV model but fast microbial adjustments.

REV Model with maintenance respiration
As REV model but maintenance respiration added.

OPT Model
Optimisation of microbial enzyme production to maximise microbial growth, and consideration of carbon costs associated with enzyme synthesis.

OPT Model with equilibrium microbes
As OPT model but fast microbial adjustments.

OPT Model with maintenance respiration
As OPT model but maintenance respiration added.

FOD Model
First order decomposition model, modified to account for temperature sensitive carbon use efficiency. Combined cost and the half saturation constants at μ = 0, 0.1, and 0.5, respectively.

FOD Model
k* hr -1 1.71*10 -5 First order decay constant @ 15°C This study * k in FOD model is identical to V max,OPT in OPT model. respiration to a 5°C warming in the base models (FWD vs REV and OPT, Fig. 1a). The black 20 line represents initial values, which are model equilibria at15°C. We chose logarithmic axes for 21 time to better highlight the differences in short-term responses. We note that the differences in 22 simulated soil organic carbon and respiration for the OPT and the FOD are almost equal and therefore not discernible. Also, values of CUE at warmed temperature are identical in all models, 1 and therefore, the orange line is superimposed on blue and green lines. In the OPT model, 2 simulations are carried out at zero enzyme production cost, i.e. μ 2 = Kp*c = 0).